This document discusses properties and theorems related to perpendicular lines: - Two lines are perpendicular if the product of their slopes is -1. Vertical and horizontal lines are also perpendicular. - If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. - If two lines are each perpendicular to the same line, then they are parallel to each other. - It provides a proof that if two coplanar lines are each perpendicular to the same line, then they are parallel to each other. - Examples are given to determine if given lines are perpendicular or parallel based on their slopes or equations. Homework exercises are assigned from the textbook.

1. Proving Lines are Perpendicular

2. Properties of Perpendicular Lines
Perpendicular Lines Postulate:
• l1⊥l2 if and only if
m1∙m2 = -1
• That is, m2 = -1/m1,
The slopes are
negative reciprocals
of each other.
• Two non-vertical lines are perpendicular if and
only if the product of their slopes is -1.
Vertical and horizontal lines are perpendicular.

3. • In a plane, if a line is perpendicular to
one of two parallel lines, then it is
perpendicular to the other.
Theorem: Perpendicular to Parallel Lines:
and
Then

4. • If two coplanar lines are each
perpendicular to the same line, then
they are parallel to each other.
Theorem: Two Perpendiculars:

5. Proof of Perpendicular to Parallel Lines Theorem
Statement Reason
1 l ll m, l ⊥ n Given
2 ∠1 is a right angle Definition of lines⊥
3
m∠1 = 90o
Definition of a right angle
4
m 2∠ = m∠1
Corresponding angles postulate
5
m∠2 = 90o
Substitution property of equality
6 ∠2 is a right angle Definition of a right angle
7 m ⊥ n Definition of lines⊥
Given: l ll m and l ⊥ n
Prove: m ⊥ n

6. Examples
1. Line r contains the points (-2,2) and (5,8).
Line s contains the points (-8,7) and (-2,0).
Is r ⊥ s?

7. 2. Given the equation of line v is
and line w is
Is v ⊥ w?

8. Given the line
3.Find the equation of the line passing through (
6,1) and perpendicular to the given line.
4. Find the equation of the line passing through
( 6,1) and parallel to the given line.

9. Homework
• Exercise 3.7 page 175: 1-35, odd.

10. Homework
• Exercise 3.7 page 175: 1-35, odd.